We report on radio observations of the M8.5 dwarf LSR J1835+3259 and the L3.5 dwarf 2MASS J00361617+ 1821104, which provide the strongest evidence to date that the electron cyclotron maser instability is the dominant mechanism producing radio emission in the magnetospheres of ultracool dwarfs. As has previously been reported for the M9 dwarf TVLM 513-6546, periodic pulses of 100% circularly polarized, coherent radio emission are detected from both dwarfs with periods of 2.84 +/- 0.01 and 3.08 +/- 0.05 hr, respectively, for LSR J1835+3259 and 2MASS J00361617+ 1821104. Importantly, periodic unpolarized radio emission is also detected from 2MASS J00361617+ 1821104, and brightness temperature limitations rule out gyrosynchrotron radiation as a source of this radio emission. The unpolarized emission from this and other ultracool dwarfs is also attributed to electron cyclotron maser emission, which has become depolarized on traversing the ultracool dwarf magnetosphere, possibly due to propagations effects such as scattering. Based on available v sin i data in the literature and rotation periods derived from the periodic radio data for the three confirmed sources of electron cyclotron maser emission, TVLM 513-46546, LSR J1835+3259, and 2MASS J00361617+ 1821104, we determine that the rotation axes of all three dwarfs are close to perpendicular to our line of sight. This suggests a possible geometrical selection effect due to the inherent directivity of electron cyclotron maser emission, that may account for the previously reported relationship between radio activity and v sin i observed for ultracool dwarfs. We also determine the radius of the dwarf LSR J1835+3259 to be >= 0.117 +/- 0.012 R-circle dot. The implied size of the radius, together with the bolometric luminosity of the dwarf, suggests that either LSR J1835 is a young- or intermediate-age brown dwarf, or that current theoretical models underestimate the radii of ultracool dwarfs.